Rough paths methods 1: Introduction Samy Tindel University of Lorraine at Nancy KU - Probability Seminar - 2013 Samy T. (Nancy) Rough Paths 1 KU 2013 1 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 2 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 3 / 44
Nancy 1 Nancy, France - Google Maps 04/09/13 05:45 Address Nancy Samy T. (Nancy) Rough Paths 1 KU 2013 4 / 44
Nancy 2: Stanislas square Samy T. (Nancy) Rough Paths 1 KU 2013 5 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 6 / 44
Equation under consideration Equation: Standard differential equation driven by fBm, R n -valued � t � t d � 0 V j ( Y s ) dB j Y t = a + 0 V 0 ( Y s ) ds + s , (1) j = 1 with t ∈ [ 0 , 1 ] . Vector fields V 0 , . . . , V d in C ∞ b . A d -dimensional fBm B with 1 / 3 < H < 1. Note: some results will be extended to H > 1 / 4. Samy T. (Nancy) Rough Paths 1 KU 2013 7 / 44
Fractional Brownian motion B = ( B 1 , . . . , B d ) B j centered Gaussian process, independence of coordinates Variance of the increments: E [ | B j t − B j s | 2 ] = | t − s | 2 H H − ≡ Hölder-continuity exponent of B If H = 1 / 2, B = Brownian motion If H � = 1 / 2 natural generalization of BM Remark: FBm widely used in applications Samy T. (Nancy) Rough Paths 1 KU 2013 8 / 44
Examples of fBm paths H = 0 . 3 H = 0 . 5 H = 0 . 7 Samy T. (Nancy) Rough Paths 1 KU 2013 9 / 44
Paths for a linear SDE driven by fBm dY t = − 0 . 5 Y t dt + 2 Y t dB t , Y 0 = 1 H = 0 . 5 H = 0 . 7 Blue: ( B t ) t ∈ [ 0 , 1 ] Red: ( Y t ) t ∈ [ 0 , 1 ] Samy T. (Nancy) Rough Paths 1 KU 2013 10 / 44
Some applications of fBm driven systems Biophysics, fluctuations of a protein: New experiments at molecule scale ֒ → Anomalous fluctuations recorded Model: Volterra equation driven by fBm ֒ → Samuel Kou Statistical estimation needed Finance: Stochastic volatility driven by fBm (Sun et al. 2008) Captures long range dependences between transactions Samy T. (Nancy) Rough Paths 1 KU 2013 11 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 12 / 44
Rough paths assumptions Context: Consider a Hölder path x and For n ≥ 1, x n ≡ linearization of x with mesh 1 / n → x n piecewise linear. ֒ For 0 ≤ s < t ≤ 1, set � x 2 , n , i , j s < u < v < t dx n , i u dx n , j ≡ st v Rough paths assumption 1: x is a C γ function with γ > 1 / 3. The process x 2 , n converges to a process x 2 as n → ∞ → in a C 2 γ space. ֒ Rough paths assumption 2: Vector fields V 0 , . . . , V j in C ∞ b . Samy T. (Nancy) Rough Paths 1 KU 2013 13 / 44
Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n � 0 V j ( y n u ) dx n , j t = a + u ) du + u . j = 1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d → Y t = a + 0 V j ( Y u ) dx j ֒ u . j = 1 Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44
Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n � 0 V j ( y n u ) dx n , j t = a + u ) du + u . j = 1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d → Y t = a + 0 V j ( Y u ) dx j ֒ u . j = 1 Rough paths theory Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44
Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n � 0 V j ( y n u ) dx n , j t = a + u ) du + u . j = 1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d → Y t = a + 0 V j ( Y u ) dx j ֒ u . j = 1 � dx , � � dxdx Rough paths theory Smooth V 0 , . . . , V d Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44
Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n � 0 V j ( y n u ) dx n , j t = a + u ) du + u . j = 1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d → Y t = a + 0 V j ( Y u ) dx j ֒ u . j = 1 � dx , � � dxdx � V j ( x ) dx j Rough paths theory dy = V j ( y ) dx j Smooth V 0 , . . . , V d Samy T. (Nancy) Rough Paths 1 KU 2013 14 / 44
Iterated integrals and fBm Nice situation: H > 1 / 4 → 2 possible constructions for geometric iterated integrals of B . ֒ Malliavin calculus tools ֒ → Ferreiro-Utzet Regularization or linearization of the fBm path ֒ → Coutin-Qian, Friz-Gess-Gulisashvili-Riedel Conclusion: for H > 1 / 4, one can solve equation dY t = V 0 ( Y t ) dt + V j ( Y t ) dB j t , in the rough paths sense. Remark: Recent extensions to H ≤ 1 / 4 (Unterberger, Nualart-T). Samy T. (Nancy) Rough Paths 1 KU 2013 15 / 44
Study of equations driven by fBm Basic properties: Moments of the solution 1 Continuity w.r.t initial condition, noise 2 More advanced natural problems: Density estimates 1 ֒ → Hu-Nualart + Lots of people 0.16 0.14 Numerical schemes 2 0.12 ֒ → Neuenkirch-T, Friz-Riedel 0.1 0.08 Invariant measures, ergodicity 0.06 3 0.04 ֒ → Hairer-Pillai, Cohen-Panloup-T 0.02 0 −10 −5 0 5 10 Statistical estimation ( H , coeff. V j ) 4 ֒ → Berzin-León, Hu-Nualart, Neuenkirch-T Samy T. (Nancy) Rough Paths 1 KU 2013 16 / 44
Extensions of the rough paths formalism Stochastic PDEs: Equation: ∂ t Y t ( ξ ) = ∆ Y t ( ξ ) + σ ( Y t ( ξ )) ˙ x t ( ξ ) ( t , ξ ) ∈ [ 0 , 1 ] × R d Easiest case: x finite-dimensional noise Methods: → viscosity solutions or adaptation of rough paths methods ֒ KPZ equation: Equation: ∂ t Y t ( ξ ) = ∆ Y t ( ξ ) + ( ∂ ξ Y t ( ξ )) 2 + ˙ x t ( ξ ) − ∞ ( t , ξ ) ∈ [ 0 , 1 ] × R x ≡ space-time white noise ˙ Methods: ◮ Extension of rough paths to define ( ∂ x Y t ( ξ )) 2 ◮ Renormalization techniques to remove ∞ Samy T. (Nancy) Rough Paths 1 KU 2013 17 / 44
Aim Define rigorously an equation driven by fBm or general Gaussian 1 processes Solve this kind of equation 2 Investigate some properties of the solution 3 ◮ Law of X t ◮ Statistical issues ◮ Other systems (delayed equations, stochastic PDEs) Samy T. (Nancy) Rough Paths 1 KU 2013 18 / 44
General strategy In order to reach the general case, we shall go through the 1 following steps: ◮ Itô integral for usual Brownian motion ◮ Young integral (2 versions) for H > 1 / 2 ◮ Case 1 / 3 < H < 1 / 2, with a semi-pathwise method For each case, 2 main steps: 2 � u s dB s ◮ Definition of a stochastic integral for a reasonable class of processes u ◮ Resolution of the equation by means of a fixed point method Samy T. (Nancy) Rough Paths 1 KU 2013 19 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 20 / 44
Sketch Introduction 1 Motivations for rough paths techniques Summary of rough paths theory Usual Brownian case 2 Basic properties of Brownian motion Itô’s stochastic integral Stochastic differential equations Samy T. (Nancy) Rough Paths 1 KU 2013 21 / 44
Definition of Brownian motion Complete probability space: (Ω , F , ( F t ) t ≥ 0 , P ) Definition 1. One-dimensional Brownian motion: continuous adapted ( B t ∈ F t ) process such that B 0 = 0 and: If 0 ≤ s < t , B t − B s independent of F s ( B t − B s ) ∼ N ( 0 , t − s ) d -dimensional Brownian motion: B = ( B 1 , . . . , B d ) , where B i independent 1-d Brownian motions Samy T. (Nancy) Rough Paths 1 KU 2013 22 / 44
Kolmogorov criterion Notation: If f : [ 0 , T ] → R d is a function, we shall denote: | δ f st | δ f st = f t − f s , and � f � µ = sup | t − s | µ s , t ∈ [ 0 , T ] Theorem 2. Let X = { X t ; t ∈ [ 0 , T ] } be a process defined on (Ω , F , P ) , such that E [ | δ X st | α ] ≤ c | t − s | 1 + β , for s , t ∈ [ 0 , T ] , c , α, β > 0 Then there exists a modification ˆ X of X such that almost surely ˆ 1 for any γ < β/α , i.e P ( ω ; � ˆ X ∈ C γ X ( ω ) � γ < ∞ ) = 1. Samy T. (Nancy) Rough Paths 1 KU 2013 23 / 44
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